Sorry to be the bearer of bad news, but the BC350A has a relatively weak AM Aircraft receiver. It also tends to be relatively noisy internally. This is one of the reasons that radio is usually significantly cheaper on the used gear market than some of its older and newer cousins from Bearcat, GRE, and Radio Shack (which are also typically made by GRE).
You have a good antenna. I assume you also have some decent feed line between the antenna and the radio. How far away are you from the airport and aircraft you are listening to?
I can take various receivers here and compare them on the identical antenna system -- and the poor BC350A is the under-performer of the bunch. It works, but it's noisy.
I haven't hooked it up to the test gear to measure receiver sensitivity, but I suspect, from lots of years of tuning other receivers, that it's about .2uV deaf compared to the other radios around here.
With *some* receivers I might recommend a powered RF pre-amplifier ahead of the radio if you're not in an area with tons of other RF sources (as those will tend to overload your receiver's front-end filters of which most scanners have very little since they're designed to hear large swaths of RF spectrum frequency-wise), but I don't think it can significantly help the BC350A.
Save up some sheckles for a little better radio. It's worth it.